Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,21)(15,22,19,16)(17,25,26,20)(23,24,27,28)(30,34)(31,37)(32,39)(43,44);(6,7)(10,11)(15,19)(16,22)(23,27)(24,28)(29,30,33,34)(31,35,37,41)(32,36,39,42)(38,43,40,44);(5,6,9,7)(8,11,12,10)(13,15,18,19)(14,16,21,22)(17,23,26,27)(20,24,25,28)(29,31)(30,35)(32,40)(33,37)(34,41)(36,44)(38,39)(42,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16)(14,19)(15,21)(17,24)(18,22)(20,27)(23,25)(26,28)(29,30)(31,35)(32,36)(33,34)(37,41)(38,43)(39,42)(40,44);(5,8)(6,10)(7,11)(9,12)(13,17,18,26)(14,20,21,25)(15,23,19,27)(16,24,22,28)(29,32)(30,36)(31,38)(33,39)(34,42)(35,43)(37,40)(41,44)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=h,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4509397);